Conditioned local limit theorems for random walks defined on finite Markov chains

TL;DR

This paper develops conditioned local limit theorems for sums of functions along finite Markov chains, analyzing the asymptotic probabilities of certain events related to the chain's partial sums and first hitting times.

Contribution

It introduces a conditioned local limit theorem for Markov chains and provides asymptotic estimates for probabilities involving the chain's partial sums and hitting times.

Findings

Established a conditional local limit theorem for Markov chains.

Derived asymptotic order of probabilities as n→∞.

Generalized results for applications in stochastic processes.

Abstract

Let $(X_n)_{n\geq 0}$ be a Markov chain with values in a finite state space $\mathbb X$ starting at $X_0=x \in \mathbb X$ and let $f$ be a real function defined on $\mathbb X$. Set $S_n=\sum_{k=1}^{n} f(X_k)$, $n\geqslant 1$. For any $y \in \mathbb R$ denote by $\tau_y$ the first time when $y+S_n$ becomes non-positive. We study the asymptotic behaviour of the probability $\mathbb P_x \left( y+S_{n} \in [z,z+a] \,,\, \tau_y > n \right)$ as $n\to+\infty.$ We first establish for this probability a conditional version of the local limit theorem of Stone. Then we find for it an asymptotic equivalent of order $n^{3/2}$ and give a generalization which is useful in applications. We also describe the asymptotic behaviour of the probability $\mathbb P_x \left( \tau_y = n \right)$ as $n\to+\infty$.